### Finitely generated quadratic modules

^{∗}

### Yuriy A. Drozd

Yuriy Drozd: Max-Plank-Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany, and Kiev Taras Shevchenko University, De- partment of Mechanics and Mathematics, Volodimirska 64, 01033 Kiev, Ukraine;

e-mail: [email protected]

Mailing address:

Popudrenko 22/14, ap.19 02100 Kiev, Ukraine

∗This research was partially supported by Grant UM1-327 of CRDF and Ukrainian Government

Abstract. A complete description of finitely generated quadratic modules is given in terms of their projective presentations as well as of generators and relations. The main tool is the reduction of this description to a sort of “matrix problem.”

1. Introduction

Polynomial functors were introduced by Whitehead [15], Eilenberg and MacLane [13], and since then they have proved their essential role in a lot of problems of algebraic topology. In particular, their simplest case, quadratic functors, is widely used in homotopy theory (cf., e.g., [15, 2, 3]). They provide natural homotopy invariants of polyhedra [2], and someties these invariants are even enough for a complete classifica- tion of homotopy types [5]. Hence, a description of such functors seems an interesting and rather important problem. Fortunately, it is indeed quite possible, at least for the case of “right continuous” functors, de- termined by their values on free abelian groups, or, the same,quadratic modules in the sense of [2, 3]. Note that till now such a description has only been given for quadratic modules of “cyclic type” [2]. It turns out that the classification of quadratic modules is a special case of the problem considered by the author in [11], hence, can be reduced to the representations of the so called “bunches of chains” in the sense of [7]. In this paper we present such a reduction and deduce from [7] a complete description of finitely generated quadratic modules. Namely, after general definitions given in Section 2, we prove in Section 3 that the classification of quadratic modules should indeed be done locally, i.e., it is enough to consider their 2-adic localizations. Section 4 is the crucial one: it relates quadratic modules to the representations of a certain bunch of chains, while in Section 5 a description of quadratic modules is given via their projective presentations. In Section 6 we rewrite it in terms of generators and relations and give some corollaries of this description: projective resolutions of quadratic modules, their torsion free parts, etc.

We add an Appendix devoted to the representations of bunches of chains, where we reformulate and rearrange a trifle the results of [7]

taking into account the specific of our case (we need not but chains, while in [7] also semi-chains are considered). The answer is given in combinatorial frames of “strings and bands,” well-known to the ex- perts in the representation theory of finite-dimensional algebras. One may suppose that the relations between this theory and some prob- lems arising from topology should be rather wide and fruitful (cf., e.g., [4, 5]).

2

2. Generalities

We remind some definitions and examples related to quadratic func- tors and quadratic modules. For the backgrounds we refer to the book [3]. We denote by Ab (resp., ab) the category of abelian groups (resp., that of finitely generated ones).

A quadratic functor is a functor F : Ab → Ab such that, for ev-
ery two objects A, B, the function ˜F : Hom(A, B)×Hom(A, B) →
Hom(A, B) defined as ˜F(a, b) = F(a +b) − F(a) − F(b) is bilin-
ear. In this case, this function can be prolonged to a bilinear func-
tor ˜F : Ab → Ab such that, for every A, B, one has: F(A⊕B) '
F(A)⊕F(B)⊕F˜(A, B) . Let F_{1} =F(Z) , F_{2} = ˜F(Z,Z) ; H :F_{1} →F_{2}
be the composition F(Z)→F(Z⊕Z)→F˜(Z,Z) , where the first map-
ping is induced by the diagonal embedding Z→Z⊕Z, while the second
one is the projection onto the direct summand; at last, P : F_{2} → F_{1}
be the composition ˜F(Z,Z) → F(Z ⊕Z) → F(Z) , where the first
mapping is the embedding of the direct summand, while the second
one is induced by the addition mapping Z⊕Z → Z. Then one has:

P HP = 2P and HP H = 2H, whence the following definition:

Definition 2.1. Aquadratic Z-module(or simply, aquadratic module)
is a quadruple M = (M_{1}, M_{2}, H, P) , where M_{1}, M_{2} are abelian groups
and H : M_{1} → M_{2}, P : M_{2} → M_{1} are homomorphisms such that
P HP = 2P, HP H = 2H.

Conversely, each quadratic module M defines a quadratic functor
M, called thequadratic tensor product: the group MA is generated
by the symbols ma and n[a, b] , where m ∈M_{1}, n∈M_{2}, a, b∈A,
subject to the following relations:

m(a+b) = ma+mb+Hm[a, b],
(m+m^{0})a=ma+m^{0}a ,

n[a, b] is 3-linear, n[a, a] =P(n)a ,

n[a, b] = [b, a](HP −1)a .

It is known [6] that the quadratic tensor product is right continuous, i.e., commutes with cokernels and direct limits, and every right contin- uous quadratic functor is isomorphic to M for a quadratic module M (determined up to isomorphism). Thus, the category of quadratic modules is equivalent to that of right continuous quadratic functors.

Remark. Certainly, a right continuous functor is completely defined by its values on the full subcategory fab of finitely generated free

3

abelian groups. Thus, the category of quadratic modules is equivalent
to that of quadratic functors fab→Ab. Just in the same way, one can
identify the category of contravariant quadratic functors fab^{◦} → Ab
(or, equivalently, that of contravariant left continuous quadratic func-
tors Ab^{◦} →Ab) with the category dual to that of quadratic modules:

a quadratic module M corresponds to the “quadratic Hom-functor” HOM( , M) [3, Definition 6.13.14].

Quadratic modules can be considered as modules over a special ring A, for which we need a more explicit construction than that in [3].

Namely, we define A as the subring of the direct product Z×Z× Mat(2,Z) consisting of all triples

a, b,

c_{1} 2c_{2}
c_{3} c_{4}

such that a≡c1 (mod 2) and b ≡c4 (mod 2). Let

e_{1} =

1,0, 1 0

0 0

, e_{2} =

0,1, 0 0

0 1

, h=

0,0,

0 0 1 0

, p=

0,0, 0 2

0 0

.

To each A-module M one associates the quadratic module (e_{1}M, e_{2}M,
h_{M}, p_{M}) , where h_{M} and p_{M} denote, respectively, the multiplication
by h and by p in the module M. Conversely, each quadratic module
(M_{1}, M_{2}, H, P) gives rise to an A-module M such that M =M_{1}⊕M_{2}
as a group and

a, b,

a+ 2c_{1} 2c_{2}
c_{3} b+ 2c_{4}

m_{1}
m_{2}

=

am_{1}+c_{2}P m_{2}+c_{1}P Hm_{1}
bm_{2}+c_{3}Hm_{1}+c_{4}HP m_{2}

.
Hence, the category QM of quadratic modules is equivalent to the
category A-Mod of (left) A-modules. Moreover, this correspondence
maps finitely generated A-modules to finitely generated quadratic mod-
ules (i.e., such that both M_{1} and M_{2} are finitely generated groups)
and vice versa. The advantage of this realization of the ring A is that
it is anorder in the semi-simple algebra Q×Q×Mat(2,Q) , so we can
apply the general theory of such orders to study quadratic modules.

Note that the natural involution δ of the category of quadratic mod-
ules that maps (M_{1}, M_{2}, H, P) to (M_{2}, M_{1}, P, H) corresponds to the
following automorphism of the ring A:

a, b,

c1 2c2

c_{3} c_{4}

7→

b, a,

c4 2c3

c_{2} c_{1}

.

4

This involution induces an involution on the category of right continu- ous quadratic functors, which we also denote by δ.

Here are some examples of quadratic functors and the corresponding A-modules, which play an important role in what follows:

Examples. (1) The tensor square ⊗^{2} : A 7→ A⊗A corresponds
to the projective A-module A_{2} = Ae_{2}. Its dual P^{2} =δ⊗^{2} is
the so-called “quadratic construction” mapping a group A to
I(A)/I^{3}(A) , where I(A) is the augmentation ideal of the group
ring Z[A] . It corresponds to the projective module A_{1} =Ae_{1}.
By the way, A_{1} and A_{2} are the only indecomposable projective
A-modules.

(2) Denote by B1 the projection of A1 onto Z×0×0 , by B2 the
projection of A_{2} onto 0×Z×0 , by B_{3} and B_{4}, respectively,
the projections of A_{1} and of A_{2} onto 0×0×Mat(2,Z) . Then
B_{2} and B_{4} correspond, respectively, to the functors of theouter
square V2

and of the symmetric square S^{2}, while B_{1} and B_{3}
correspond to their duals, δV2

= I being the identity functor
and δS^{2} = Γ^{2} being the Whitehead functor, which represents
the “universal quadratic function” [15].

Consider the subring B of Z×Z×Mat(2,Z) consisting of all triples

a, b,

c_{1} 2c_{2}
c_{3} c_{4}

, and its ideal J consisting of the triples

2a,2b,

2c_{1} 2c_{2}
c_{3} 2c_{4}

.

This ideal is indeed the conductor of B in A, i.e., the biggest ideal
of B contained in A. The ring B ishereditary(i.e., of the global ho-
mological dimension 1). Hence, each finitely generated B-module is a
direct sum of factor-modules N/N^{0}, with N being an indecomposable
projective B-module. In our case, N is isomorphic to one of the mod-
ules B_{j} defined above. Moreover, every homomorphism g : Q → P
of projective (finitely generated) B-modules is “diagonalizable,” i.e.,
there are decompositions of Q and P into direct sums of indecompos-
able modules: Q=Lm

i=1Q_{i} and P =Ln

j=1P_{j}, such that the matrix
of g with respect to this decomposition is diagonal [10].

3. Localization

From now on all modules are supposedfinitely generated, so we omit these words everywhere. In terms of quadratic functors, we restrict

5

ourselves by the functors fab→ab. For any abelian group (in partic- ular, for every A-module) M denote by tM its torsion part, i.e., the set of all elements of finite order, and by fM itstorsion free part i.e.

the factor M/tM.

For any prime p∈ N, denote by _{(p)} the functor of the (complete)
localization at p. It maps every abelian group A to A_{(p)} =A⊗Zp,
where Zp is the ring of p-adic integers. Evidently, A_{(p)} =B_{(p)} =Zp×
Zp ×Mat(2,Zp) if p 6= 2 . Hence, in this case, every indecomposable
A_{(p)}-module is isomorphic either to one of the modules B_{i(p)} (i =
1,2,3 ) or to a factor B_{i(p)}/p^{k}B_{i(p)} for some k. (Note that, for p6= 2 ,
B_{3(p)} 'B_{4(p)}.)

Put Q=Q×Q×Mat(2,Q) . For every A-module M, Q⊗_{A}M is a
module over the semi-simple algebra Q, hence, Q⊗_{A}M 'L3

i=1r_{i}U_{i},
where U_{i} are simple Q-modules: U_{i} =Q⊗_{A}B_{i} (note that U_{4} 'U_{3}).

Denote by r(M) the vector (r_{1}, r_{2}, r_{3}) , which we call the rank vector
of the module M. We use the same definitions and notations for A_{(p)}-
modules, certainly, replacing Q by the field Qp of p-adic numbers.

Theorem 3.1. (1) A-modules M, N are isomorphic if and only
if M_{(p)} 'N_{(p)} for all prime p.

(2) Given any set {Np}, where Np is an A(p)-module, there is
an A-module M such that M_{(p)} ' N_{p} for all p if and only
if almost all (i.e., all but a finite set) of them are torsion free
(maybe, zero) and r(N_{p}) = r(N_{q}) for all p, q. In this case,
r(M) =r(N_{p}) and tM =L

ptN_{p}.

Proof. The theorem is obvious if all considered modules are torsion.

Consider the case when these modules aretorsion free. Then the second
claim of the theorem is well-known [14]. The first one is also obvious for
torsion free B-modules, as they are direct sums of B_{i} and EndB_{i} =Z
(cf., e.g., [8]). Moreover, given any torsion free A-module M, the
isomorphism classes of modules N such that N_{(p)} ' M_{(p)} for all p
and B⊗_{A} N ' B ⊗_{A}M are in one-to-one correspondence with the
double cosets

(1) Aut(B⊗_{A}M)\Aut(B⊗_{A}M_{(2)})/AutM_{(2)}

(cf. ibid.). But, certainly, Hom_{A}(M, M^{0})⊇Hom_{B}(B⊗_{A}M, M^{0}J) for
every two A-modules M, M^{0} and all the factors

Hom_{B}(B_{i},B_{j})/Hom_{B}(B_{i}, JB_{j})

are zero for i 6= j and F^{2} for i = j. Hence, any automorphism
of B⊗_{A}M_{(2)} is congruent to an automorphism of B⊗_{A} M modulo

6

AutM_{(2)}. Thus, the set (1) always consists of a unique element, so,
the theorem is true for torsion free A-modules.

For any A-module M, one can consider the exact sequence 0−→tM −→M −→fM −→0

which defines an element ξ_{M} ∈Ext^{1}_{A}(fM,tM) . As there are no homo-
morphisms from tM to fM, two modules N and M are isomorphic
if and only if there are isomorphisms α : tM →tN and β : fM →fN
such that αξ_{M} = ξ_{N}β (we mean the Yoneda multiplication). Put
F = fM and T = tM. Then Ext^{1}_{A}(F, T) ' Ext_{A}_{(2)}(F_{(2)}, T_{(2)}) (as
A_{(p)} is hereditary for p6= 2 ). In particular, one only has to consider
the case, when T is a 2-group, i.e., T =T_{(2)}. Thus, the second claim
of the theorem becomes evident. Moreover, in this case the isomor-
phism classes of the A-modules N such that N_{(p)} ' M_{(p)} for all p
are in one-to-one correspondence with the double cosets

(2) AutF \AutF_{(2)}/Aut_{0}F_{(2)},

where Aut_{0}F_{(2)} denotes the subgroup of all automorphism acting triv-
ially on Ext^{1}_{A}(F, T) .

Now note that J_{(2)} = radA_{(2)} (Jacobson radical) and the ring A_{(2)}
is semi-perfect in the sense of [1]. Hence, there always is an exact
sequence

0−→K −→P −→F_{(2)} −→0,

where P is a projective A_{(2)}-module and K ⊆ J P . It induces an
exact sequence

0−→K −→J P −→J F_{(2)} −→0.

As both J P and J F_{(2)} are B-modules and B is hereditary, the latter
splits. It implies that αξ = 0 for any endomorphism α : F_{(2)} → F_{(2)}
whose image belongs to J F_{(2)} and for any element ξ ∈ Ext^{1}_{A}(F, T) .
Hence, Aut_{0}F_{(2)} contains the “congruence subgroup”

α∈AutF_{(2)}| Im(α−id) ⊆J F_{(2)} .

One can easily check that B_{i} are the only irreducible torsion free
A-modules. As each of them is a factor-module of some A_{j} and the
length of Q⊗_{A}A_{j} is 2 , A_{(2)} is an order of weight 2 in the sense of
[12]. Moreover, as JAj decomposes, it also follows from [12] that the
modules Aj are the only indecomposable but not irreducible torsion
free A-modules. Thus, every torsion free A-module is a direct sum of
modules isomorphic to B_{i} or A_{j}. The next table describes the values

7

of the groups HomA_{(2)}(F_{(2)}^{0} , F_{(2)}^{00} )/HomA_{(2)}(F_{(2)}^{0} , J F_{(2)}^{00} ) , when F^{0} and
F^{00} run through indecomposable torsion free A-modules:

B_{1} B_{2} B_{3} B_{4} A_{1} A_{2}
B_{1} F2 0 0 0 F2 0
B_{2} 0 F2 0 0 0 F2

B3 0 0 F^{2} 0 F^{2} 0
B4 0 0 0 F^{2} 0 F^{2}
A_{1} 0 0 0 0 F^{2} 0
A_{2} 0 0 0 0 0 F^{2}

It evidently implies that every automorphism of F_{(2)} is congruent to
an automorphism of F modulo Aut0F_{(2)}, i.e., the set (2) consists of
one element. Hence, the theorem is valid for all (finitely generated)

A-modules.

4. Reduction

From now on, except of the last section (“Corollaries”), we con-
sider the 2-local case, hence, always write A, B, Q, etc. instead of
A_{(2)}, B_{(2)}, Q_{(2)}, etc.

We shall studyprojective presentationsofA-modulesM, i.e., homo-
morphisms of projective modules f :P^{0} →P such that M ' Cokerf.

In terms of quadratic functors, it means that we shall study exact se- quences

Φ^{0} →Φ→F →0,

where F is a given functor, while Φ and Φ^{0} are direct sums of func-
tors isomorphic to ⊗^{2} and P^{2}. Moreover, if we are studying finitely
generated modules, these sums can always be chosen finite too. The
advantage of the local case is that here A is asemi-perfect ring in the
sense of [1]; hence, every (finitely generated) A-module M has apro-
jective cover, i.e., an epimorphism ϕ:P →M, where P is projective
and Kerϕ ⊆J P (note that here J = radA, the Jacobson radical of
A). It implies that M has a projective presentation f :P^{0} →P with
Imf ⊆J P and we only consider such presentations.

Each homomorphism of projective modules f :P^{0} → P induces the
homomorphism B⊗f : B⊗_{A} P^{0} → B⊗_{A} P. If M = Cokerf, then
Coker(B⊗ f) ' B ⊗_{A} M. As we know all possibilities for B ⊗ f,
we are going to classify the homomorphisms f with prescribedB⊗f,
or, the same, the A-modules M with prescribed B⊗AM. We always
identify P with the submodule 1⊗P ⊆ B⊗_{A}P. If Imf ⊆ J P, then
also Im(B⊗f)⊆J(B⊗_{A}P) =J P. On the other hand, suppose that

8

Imϕ⊆J P for someϕ:B⊗AP^{0} →B⊗AP. As HomB(B⊗AP^{0}, J P)'
Hom_{A}(P^{0}, J P), then ϕ is of the formB⊗f for some f :P^{0} →P.

One calls two homomorphisms f, f^{0} : P^{0} → P equivalent if there
are automorphisms g ∈ AutP and g^{0} ∈ AutP^{0} such that gf = f^{0}g^{0}.
Then, of course, Cokerf 'Cokerf^{0}. The following proposition is quite
evident:

Proposition 4.1. LetP, P^{0} be projective (finitely generated)A-modules,
Q = B⊗_{A} P, Q^{0} = B ⊗_{A} P^{0}, ϕ : Q^{0} → Q a homomorphism with
Imϕ⊆J Q. There is a one-to-one correspondence between the equiva-
lence classes of homomorphisms f :P^{0} →P such that B⊗f =ϕ and
the double cosets

(3) Autϕ\AutQ×AutQ^{0}/AutP ×AutP^{0},

whereAutϕdenotes the subgroup{(g, h)∈AutQ×AutQ^{0}|gϕ=ϕh}.

Namely, the coset containing a pair (g, h) corresponds to the homo-
morphism f which coincides with g^{−1}ϕh under the identification of
Hom_{A}(P^{0}, J P) and Hom_{B}(Q^{0}, J Q) .

Put P = P/J P and Q = Q/J Q. Note that always J Q = J P and AutP contains the “congruence subgroup”

Aut(Q, J) ={g ∈AutQ| Im(g−id) ⊆J Q} .

Moreover, an endomorphism g of Q is invertible if and only if it is
invertible modulo J. Therefore, the set (3) can be identified with
(4) Autϕ\AutQ×AutQ^{0}/AutP ×AutP^{0},

where Autϕ is the image of Autϕ in AutQ×AutQ^{0}.

Every homomorphism ϕ:Q^{0} →Q of projective B-modules is equiv-
alent to a direct sum of homomorphisms of the forms ϕijk : Bj → Bi.
Here (ij)∈ {(11),(22),(33),(34),(43),(44)},k is a non-negative inte-
ger or, if i=j, maybe k =±∞. If k 6=±∞, the homomorphism ϕ_{ijk}
is the multiplication by 2^{k+1} for i =j, the multiplication on the right
by 2^{k}h for (ij) = (34) and by 2^{k}p for (ij) = (43); ϕii,−∞ denotes the
homomorphism 0→Pi and ϕii,+∞ the homomorphism Pi →0.

The rings A/J and B/J are semi-simple; A_{i} (i= 1,2) and B_{j} (j =
1,2,3,4) are their simple modules. Moreover, B_{1} ' B_{3} ' A_{1} and
B_{2} 'B_{4} 'A_{2} asA-modules, and the isomorphismsB⊗_{A}A_{1} 'B_{1}⊕B_{3},
B⊗_{A}A_{2} 'B_{2}⊕B_{4} induce the diagonal embeddings

EndA_{1} 'F2 −→ EndB_{1} ×EndB_{3} 'F^{2}2,
EndA_{2} 'F2 −→ EndB_{2} ×EndB_{4} 'F^{2}2,

9

Forϕ :Q^{0} →Q,ψ :Q^{0}_{1} →Q1, one denotes:

Hom(ϕ, ψ) = {(u, v)|u:Q→Q_{1}, v :Q^{0} →Q^{0}_{1}, uϕ=ψv},
H(ϕ, ψ) and H^{0}(ϕ, ψ) the projections of Hom(ϕ, ψ), respectively, onto
Hom_{B}(Q, Q_{1}) and Hom_{B}(Q^{0}, Q^{0}_{1}). One easily calculates these groups
whenϕ, ψrun through{ϕ_{ijk}}. It is convenient to present the result in
the following way. Consider the new symbolsv(ijk) andv^{0}(ijk), where
the triples (ijk) are as before except v(ii,+∞) and v^{0}(ii,−∞), which
are forbidden. We define an ordering on the set of these symbols putting
v(ijk) ≤ v(i^{0}j^{0}k^{0}) if H(ϕ(ijk), ϕ(i^{0}j^{0}k^{0}))6= 0 and v^{0}(ijk) ≤ v^{0}(i^{0}j^{0}k^{0}) if
H^{0}(ϕ(ijk), ϕ(i^{0}j^{0}k^{0}))6= 0 (then all these groups are isomorphic to F2).

Here is the table describing this ordering:

v(ijk)< v(ijk^{0}) ifk^{0} < k ,

v(33k)< v(34k^{0}) if k^{0} < k , otherwise v(34k^{0})< v(33k),
v(44k)< v(43k^{0}) if k^{0} ≤k , otherwise v(43k^{0})< v(44k),
v^{0}(ijk)< v^{0}(ijk^{0}) if k < k^{0},

v^{0}(33k)< v^{0}(43k^{0}) ifk < k^{0}, otherwise v^{0}(43k^{0})< v^{0}(33k),
v^{0}(44k)< v^{0}(34k^{0}) ifk ≤k^{0}, otherwise v^{0}(34k^{0})< v^{0}(44k).
Now one can easily see that we have obtained a special case of the
so called “bunch of chains” (cf. [7] or Appendix). Namely, put I =
{1,2,3,4,5,6,7,8}, consider the following chains:

E1 ={v(11k)} F1 ={1}
E_{2} ={v(22k)} F_{2} ={2}
E_{3} ={v(33k), v(34k)} F_{3} ={3}
E_{4} ={v(43k), v(44k)} F_{4} ={4}
E_{5} ={v^{0}(11k)} F_{5} ={1^{0}}
E_{6} ={v^{0}(22k)} F_{6} ={2^{0}}
E_{7} ={v^{0}(33k), v^{0}(43k)} F_{7} ={3^{0}}
E_{8} ={v^{0}(34k), v^{0}(44k)} F_{8} ={4^{0}}

(with the above defined ordering) and define an equivalence relation ∼
on the union of these chains such that the only non-trivial equivalences
are: v(ijk)∼v^{0}(ijk) for k6=∞, 1∼3 , 2∼4 , 1^{0} ∼3^{0}, 2^{0} ∼4^{0}. One
gets in this way a bunch of chainsX={I,E_{i},F_{i},∼ }. We are going to
establish relations between representations of this bunch over the field
F^{2} and quadratic modules. Thus, in all references to [7] or Appendix
one supposes k = F^{2}. We write x−y if x ∈ Ei, y ∈ Fi (with the
same i) or vice versa and denote by U the bimodule corresponding to
the bunch X (cf. Apendix).

10

For every equivalence class a of the relation ∼ or, the same, for an object of the category C=C(X) (cf. Appendix), put

QE(a) =

(B_{i} if a3v(ijk) for some j, k
0 otherwise

Q^{0}_{E}(a) =

(B_{j} if a3v(ijk) for some i, k
0 otherwise

QF(a) =

(Bi⊕Bj if a ={i, j} ⊆ {1,2,3,4}

0 otherwise

Q^{0}_{F}(a) =

(B_{i}⊕B_{j} if a ={i^{0}, j^{0}} ⊆ {1^{0},2^{0},3^{0},4^{0}}

0 otherwise

If c = ⊕_{m}a_{m} is an object of the category C^{∞} (the additive hull of
C), put Q_{E}(c) = L

mQ_{E}(a_{m}) , the same for Q^{0}_{E}(c), Q_{F}(c), Q^{0}_{F}(c) .
Note that U(a, b) 6= 0 if and only if a ⊆ F, b ⊆ E and there are
x∈a, y∈b such that x−y, in which case this space is 1-dimensional
and can be identified with one of the spaces Hom_{B}(Q_{F}(a), Q_{E}(b))
or Hom_{B}(Q^{0}_{F}(a), Q^{0}_{E}(b)) , namely, the non-zero one. Hence, any el-
ement u ∈ U(c, c) can be considered as a pair of homomorphisms
(α(u), α^{0}(u)) , where α(u) : Q_{F}(c) → Q_{E}(c) and α^{0}(u) : Q^{0}_{F}(c) →
Q^{0}_{E}(c) . Call such an element u (i.e., a representation of the bunch
X) balanced if both α(u) and α^{0}(u) are isomorphisms and there are
such projective A-modules P, P^{0} that Q_{F}(c)'B⊗_{A}P and Q^{0}_{F}(c)'
B⊗_{A}P^{0}. Then the following result is immediate.

Theorem 4.2. There is a one-to-one correspondence between the equiv-
alence classes of homomorphisms of projective A-modules f :P^{0} →P
such that Imf ⊆ J P and the isomorphism classes of balanced repre-
sentations of the bunch of chains X.

5. Description

Now we are able to use the results of [7] in order to describe the
projective presentations of quadratic modules (or, the same, of A-
modules). We rearrange a bit the definitions of [7] or Appendix in
order to make them more adequate to the specific structure of our con-
crete bunch of chains X as well as to the “balancedness condition” of
Theorem 4.2. One writes x∈ X and says that x is an element of X if
x is an element of one of the chains E_{i} or F_{i}. One also writes x−y if
x∈E_{i} and y∈F_{i} with the same i or vice versa.

11

Definition 5.1. (1) An X-word (or simply a word) is a finite se-
quence w = x_{1}r_{2}x_{2}r_{3}. . . r_{n}x_{n}, where x_{i} ∈ X and r_{i} ∈ { −,∼ },
satisfying the following conditions:

(a) For every i= 2, . . . , n, x_{i−1}r_{i}x_{i} in the above defined sense.

(b) For everyi= 2, . . . , n−1, r_{i} 6=r_{i+1}.

(2) An X-word w = x_{1}r_{2}x_{2}r_{3}. . . r_{n}x_{n} is called balanced if n > 1
and both x_{1} and x_{n} are of the form v(ii,−∞) or v^{0}(ii,+∞) .
(Note that in this case necessarily r_{2} = r_{n} = − and n ≡ 0
(mod 4).) Moreover, if n = 4 , we suppose that either x_{1} or
x_{4} is of the form v(ii,−∞) .

(3) AnX-word w=x_{1}r_{2}x_{2}r_{3}. . . r_{n}x_{n} is called cyclicif r_{2} =r_{n} and
x_{n}r_{1}x_{1}, where r_{1} ∈ { ∼,− } and r_{1} 6=r_{n}. For such a word, we
put x0 =xn and xn+1 =x1. (Note that in this case necessarily
n≡0 (mod 8) .)

(4) A cyclicX-word is calledaperiodic if it cannot be obtained as a
repetition of a shorter word: w6=wrwr . . . w for r ∈ { −,∼ }.
(5) A cyclic word w = x_{1}r_{2}x_{2}r_{3}. . . r_{n}x_{n} is called normalized if it

is aperiodic, r_{2} =r_{n}=∼ and x_{1} ∈F_{i} for some i.

(The last notion is not but a technical one making easier the definition of “bands” nearby.)

In [7] (cf. also Appendix) a complete description of the representa- tions of a bunch of chains was given in terms of the so calledstrings and bands. We are going to translate it to the description of A-modules via Theorem 4.2. To this purpose, we introduce some notions and notations.

Call an A-entry any word a of the form a =x ∼y, where x ∈F_{i}
for some i and x 6= y. For every A-entry a = x ∼ y define two
projective A-modules P(a) and P^{0}(a) putting:

P(a) =

A_{1} if 1∈ {x, y}
A_{2} if 2∈ {x, y}
0 otherwise
(5)

P^{0}(a) =

A_{1} if 1^{0} ∈ {x, y}
A2 if 2^{0} ∈ {x, y}
0 otherwise
(6)

For every word w, put P(w) = L

aP(a) and P^{0}(w) = L

aP^{0}(a) ,
where a runs through all subwords of w which are A-entries.

If a and a^{0} are two subwords of w which are both A-entries, one says
that they arerelated in w by the triple (ijk) if they are contained in a
subword of the form a−ω(a, a^{0})−a^{0}, where ω(a, a^{0}) =v(ijk)∼v^{0}(ijk) ,

12

or of the form a^{0}−ω(a, a^{0})−a, where ω(a, a^{0}) = v^{0}(ijk)∼v(ijk) . Note
that in both cases P(a^{0}) = P^{0}(a) = 0 , while P(a) and P^{0}(a^{0}) are non-
zero. (In particular, this relation between a and a^{0} isnot symmetric.)
We call ω(a, a^{0}) the relating subword for a, a^{0}. If w is a normalized
cyclic word, we also include the case when a = xn−3 ∼ xn−2, a^{0} =
x_{1} ∼x_{2} or vice versa and ω(a, a^{0}) =xn−1 ∼x_{n}.

Now define the homomorphisms θ(ijk) : A_{j} → A_{i}, where k >0 or
k = 0 and i6=j. Namely, θ(ijk) is the multiplication to the right by
the following element of the ring A:

2^{k+1},0,
0 0

0 0

if i=j = 1

0,2^{k+1},
0 0

0 0

if i=j = 2

0,0,

2^{k+1} 0

0 0

if i=j = 3

0,0,

0 0
0 2^{k+1}

if i=j = 4

0,0,

0 0
2^{k} 0

if i= 3, j = 4

0,0,

0 2^{k+1}

0 0

if i= 4, j = 3

Definition 5.2. Let w = x_{1}r_{2}x_{2}r_{3}. . . r_{n}x_{n} be a balanced X-word.

Then the corresponding string homomorphism c(w) : P^{0}(w) → P(w)
is defined as one whose only non-zero components with respect to
the decompositions P(w) = L

aP(a) and P^{0}(w) = L

aP^{0}(a) are
P^{0}(a^{0}) → P(a) if a and a^{0} are related in w by a triple (ijk) , the
corresponding component being θ(ijk) .

The string module C(w) is defined as Cokerc(w) .

Note that the string homomorphism can be defined also when n = 4
and both x_{1} and x_{4} are of the form v^{0}(ii,+∞) , but then it is the zero
mapping A_{j} →0 , hence, gives rise to the zero module.

Definition 5.3. Let noww=x_{1}r_{2}x_{1}r_{2}. . . r_{n}x_{n} be a normalized cyclic
word, π(t) 6= t^{d} be a primary polynomial over the filed F2, i.e., a
power of an irreducible one, d = degπ(t) . Denote by Φ the Frobe-
nius matrix with the characteristic polynomial π(t) and by I the
unit d×d matrix. Then the band homomorphism b(w, π(t)) is de-
fined as the mapping dP^{0}(w) → dP(w) , whose only non-zero com-
ponents with respect to the decompositions dP(w) = L

adP(a) and

13

dP^{0}(w) = L

adP^{0}(a) are dP^{0}(a^{0}) → dP(a) if a and a^{0} are related
in w by a triple (ijk) , the corresponding component being θ(ijk)I,
except of the case when the relating subword ω(a, a^{0}) is the subword
x_{n−1} ∼ x_{n} (hence, {a, a^{0}} = {x_{1} ∼x_{2}, x_{n−3} ∼x_{n−2}}); in the latter
case the corresponding component is θ(ijk)Φ .

The band module B(w, π(t)) is defined as Cokerb(w, π(t)) .

The following classification is an immediate consequence of that given in [7].

Theorem 5.4. (1) For every balanced wordw, the string A-module
C(w) is indecomposable and for every pair (w, π(t)) with w a
normalized cyclic word, π(t) 6= t^{d} a primary polynomial over
F2, the band A-module B(w, π(t)) is also indecomposable.

(2) Every finitely generated A-module decomposes uniquely into a direct sum of modules isomorphic to string and band A-modules.

(3) The only isomorphisms between different string and band A- modules are the following:

(a) C(w) ' C(w^{∗}), where w^{∗} is the reversed word to w, i.e.,
if w=x_{1}r_{2}x_{2}r_{3}. . . r_{n}x_{n} then w^{∗} =x_{n}r_{n}. . . r_{3}x_{2}r_{2}x_{1}.
(b) B(w, π(t)) ' B(w^{(s)}, π(t)) where s is a positive integer

divisible by 4 and w^{(s)} is the s-shift of w, i.e., if w =
x_{1}r_{2}x_{2}. . . r_{n}x_{n}thenw^{(s)} =x_{s+1}r_{s+2}x_{s+2}. . . r_{n}x_{n}−x_{1}. . . r_{s}x_{s}.
(The condition “ 4|s” is necessary in order that w^{(s)} were
also a normalizes cyclic word.)

(c) B(w, π(t))'B(w^{∗(s−2)}, π^{∗}(t)) for s as above and π^{∗}(t) =
t^{d}π(1/t) where d= degπ(t).

6. Corollaries

We present some corollaries of the description of the A_{(2)}-modules
given above. First, note that one can also construct string and band
modules for the ring A just in the same way as it has been done in Sec-
tion 6 for its localization A_{(2)}. To be accurate, we will denote from now
on the string and band A_{(2)}-modules by C_{(2)}(w) and B_{(2)}(w, π(t)) ,
while the notations C(w) and B(w, π(t)) will be used for the string
and band A-modules. Denote also by B(i, p, k) the factor-module
B_{i}/p^{k}B_{i}, where i ∈ {1,2,3,4}, k ∈ N and p is an odd prime. As
the “local” string and band modules are indeed the 2-localizations of
the “global” ones, Theorems 3.1 and 5.4 imply the following complete
description of finitely generated A-modules (or, the same, quadratic
modules).

14

Theorem 6.1. (1) For every balanced wordw, the string A-module
C(w) is indecomposable and for every pair (w, π(t)) with w a
normalized cyclic word, π(t) 6= t^{d} a primary polynomial over
F2, the band A-module B(w, π(t)) is also indecomposable.

(2) Every finitely generated A-module decomposes uniquely into a direct sum of modules isomorphic to string and band A-modules and to modules B(i, p, k).

(3) The only isomorphisms between different string and band A- modules are the following:

(a) C(w)'C(w^{∗}).

(b) B(w, π(t)) ' B(w^{(s)}, π(t)) ' B(w^{∗(s−2)}, π^{∗}(t)) where s is
a positive integer divisible by 4.

(4) Neither two different modules B(i, p, k) are isomorphic and nei- ther of them is isomorphic to a string or band module.

We are also able to write down generators and relations for in-
decomposable quadratic modules. Namely, we say that a balanced
word w = x_{1}r_{2}x_{2}. . . r_{n}x_{n}, as well as the corresponding string mod-
ule, is of type −∞ (resp., +∞ or ±∞) if both ends of w are of
the form v(ii,−∞) (resp., both are v^{0}(ii,+∞) or one is v(ii,−∞)
and the other is v^{0}(jj,+∞) ). Certainly, in the latter case (±∞),
one can always suppose (and we shall do it) that x_{1} = v(ii,−∞)
and x_{n} = v^{0}(jj,+∞) . Just in the same way, we shall suppose that a
normalized cyclic word w starts from x1 ∈ {1,2,3,4}. Hence, such
words are of the following forms:

• word of type ±∞:

v(i_{1}i_{1},−∞)−i_{1} ∼i_{2}−v(i_{2}j_{1}k_{1})∼v^{0}(i_{2}j_{1}k_{1})−

−j_{1}^{0} ∼j_{2}^{0} −v^{0}(i_{3}j_{2}k_{2})∼ · · · ∼v^{0}(i_{2m}j2m−1k2m−1)−

−j_{2m−1}^{0} ∼j_{2m}^{0} −v^{0}(j2mj2m,+∞)

• word of type −∞:

v(i_{1}i_{1},−∞)−i_{1} ∼i_{2}−v(i_{2}j_{1}k_{1})∼v^{0}(i_{2}j_{1}k_{1})−

−j_{1}^{0} ∼j_{2}^{0} −v^{0}(i_{3}j_{2}k_{2})∼ · · · ∼v(i_{2m−1}j_{2m−2}k_{2m−2})−

−i2m−1 ∼i_{2m}−v(i_{2m}i_{2m},−∞)

• word of type +∞:

v^{0}(j−1j−1,+∞)−j_{−1}^{0} ∼j_{0}^{0} −v^{0}(i_{1}j_{0}k_{0})∼v(i_{1}j_{0}k_{0})−

−i1 ∼i2−v(i2j1k1)∼ · · · ∼v^{0}(i2mj2m−1k2m−1)−

−j_{2m−1}^{0} ∼j_{2m}^{0} −v^{0}(j_{2m}j_{2m},+∞)

15

• normalized cyclic word:

i1 ∼i2−v(i2j1k1)∼v^{0}(i2j1k1)−j_{1}^{0} ∼j_{2}^{0}−

−v^{0}(i_{3}j_{2}k_{2})∼ · · · ∼v^{0}(i_{2m−1}j_{2m−2}k_{2m−2})−j_{2m−1}^{0} ∼

∼j_{2m}^{0} −v^{0}(i_{1}j_{2m}k_{2m})∼v(i_{1}j_{2m}k_{2m})
Define the following operators acting on any quadratic module M:

R_{11}= 2−P H, R_{22}= 2−HP,

R_{33}=P H, R_{44} =HP, R_{34}=H, R_{43}=P .

Here R_{11} and R_{33} are mappings M_{1} →M_{1}, R_{22} and R_{44} are map-
pings M_{2} → M_{2}, while R_{34} : M_{2} → M_{1} and R_{43} : M_{1} → M_{2}.
(Indeed, R_{ij} corresponds to the multiplication by θ(ij0) .)

Corollary 6.2. (1) If an indecomposable quadratic module M cor-
responds to one of the balanced words described above, it is gen-
erated by the elements g_{ν} (ν = 1,2, . . . , m) such that g_{ν} ∈M_{1}
if i2ν ∈ {1,3} and gν ∈M2 if i2ν ∈ {2,4}. These elements
are subject to the following defining relations:

(7) 2^{k}^{2ν−1}R_{i}_{2ν}_{j}_{2ν−1}g_{ν} + 2^{k}^{2ν}R_{i}_{2ν+1}_{j}_{2ν}g_{ν+1} = 0.
(We put g_{0} =g_{m+1} = 0.)

(2) If an indecomposable quadratic module M corresponds to a nor-
malized cyclic word described above and to a primary polynomial
π(t) = t^{d}+a_{1}t^{d−1} +· · ·+a_{d} (a_{µ} ∈ {0,1}), it is generated by
the elements g_{νµ} (ν = 1,2, . . . , m, µ = 1,2, . . . , d) such that
g_{νµ} ∈ M_{1} if i_{2ν} ∈ {1,3} and g_{νµ} ∈ M_{2} if i_{2ν} ∈ {2,4}.
These elements are subject to the following defining relations:

2^{k}^{2ν−1}R_{i}_{2ν}_{j}_{2ν−1}g_{νµ}+ 2^{k}^{2ν}R_{i}_{2ν+1}_{j}_{2ν}g_{ν+1,µ} = 0 if ν < m ,
2^{k}^{2m−1}R_{i}_{2m}_{j}_{2m−1}g_{mµ}+ 2^{k}^{2m}R_{i}_{1}_{j}_{2m}g_{1,µ+1} = 0 if µ < d ,
2^{k}^{2m−1}R_{i}_{2m}_{j}_{2m−1}g_{md}−2^{k}^{2m}R_{i}_{1}_{j}_{2m}

d

X

µ=1

a_{µ}g_{1µ}= 0.
(8)

Remark. Certainly, the generators of M_{1} as of abelian group are, for
string modules:

{g_{ν}, phg_{ν}|g_{ν} ∈M_{1}} ∪ {pg_{ν}|g_{ν} ∈M_{2}} ,
while for band modules:

{g_{νµ}, phg_{νµ}|g_{νµ}∈M_{1}} ∪ {pg_{νµ}|g_{νµ}∈M_{2}}
and those of M2 are, for string modules:

{g_{ν}, hpg_{ν}|g_{ν} ∈M_{2}} ∪ {hg_{ν}|g_{ν} ∈M_{1}} ,

16

while for band modules:

{g_{νµ}, hpg_{νµ}|g_{νµ}∈M_{2}} ∪ {hg_{νµ}|g_{νµ}∈M_{1}} .

One can easily deduce the defining relations for these elements from the relations (7),(8): one only has to multiply each of them by h and ph or by p and hp.

Evidently, the projective indecomposable A-modules A_{1} and A_{2}
correspond to the words, respectively, v(11,−∞)−1∼3−v(33,−∞)
and v(22,−∞)−2∼4−v(44,−∞) (the only balanced words of length
4). The description given by Theorem 6.1 allows easily to calculate pro-
jective resolutions of quadratic modules. First, it implies immediately
the following result.

Proposition 6.3. The kernels of string and band morphisms are the following:

• Kerb(w, π(t)) = 0;

• Kerc(w) = 0 if both ends of the word w are of the form v(ii,−∞);

• Kerc(w) ' B_{λ(i)} if one end of the word w is v^{0}(ii,+∞) and
the other is v(jj,−∞);

• Kerc(w)'B_{λ(i)}⊕B_{λ(j)} if both ends of w are v^{0}(ii,+∞) and
v^{0}(jj,+∞),

where λ(1) = 1, λ(2) = 2, λ(3) = 4, λ(4) = 3.

Now note that each module B_{i} has a periodic projective resolution
of period 4 . Here they are:

. . .−→A_{2} −→^{α}^{1} A_{1} −→^{α}^{4} A_{1} −→^{α}^{3} A_{2} −→^{α}^{2} A_{2} −→^{α}^{1} A_{1} −→B_{1}
. . .−→A_{1} −→^{α}^{3} A_{2} −→^{α}^{2} A_{2} −→^{α}^{1} A_{1} −→^{α}^{4} A_{1} −→^{α}^{3} A_{2} −→B_{2}
. . .−→A_{1} −→^{α}^{4} A_{1} −→^{α}^{3} A_{2} −→^{α}^{2} A_{2} −→^{α}^{1} A_{1} −→^{α}^{4} A_{1} −→B_{3}
. . .−→A_{2} −→^{α}^{2} A_{2} −→^{α}^{1} A_{1} −→^{α}^{4} A_{1} −→^{α}^{3} A_{2} −→^{α}^{2} A_{2} −→B_{4}
where α_{1} =θ(341) , α_{2} =θ(221) , α_{3} =θ(431) , α_{4} =θ(111) .

Corollary 6.4. (1) A quadratic module M is of finite projective
dimension if and only if it contains no string summands of type
+∞ or ±∞. In this case either M is projective (hence, a
direct sum of modules isomorphic to A_{i}) or pr.dimM = 1.
Otherwise, pr.dimM =∞.

(2) Any quadratic module M has a periodic projective resolution

· · · →P_{n+1} −→^{γ}^{n} P_{n} −−−→^{γ}^{n−1} . . .−→^{γ}^{1} P_{1} −→^{γ}^{0} P_{0} →M →0
of period 4, namely, such that γ_{n+4} =γ_{n} for every n ≥2.

17

In particular, we obtain the value of the finitistic dimension [1] of the ring A or, the same, of the category of quadratic modules.

Corollary 6.5. fin.dimA = 1.

We are also able to calculate the torsion free part fM of every indecomposable quadratic A-module M.

Corollary 6.6. The torsion free parts of non-projective string and band modules are the following:

• fB(w, π(t)) = 0;

• fC(w) = 0 if both ends of w are of the form v^{0}(ii,+∞);

• fC(w)'Bi if one of the ends of w is v(ii,−∞) and the other
is v^{0}(jj,+∞);

• fC(w)'B_{i}⊕B_{j} if both ends of w are v(ii,−∞) and v(jj,−∞)
and w is not of length 4.

Appendix: Bunches of chains

Here we recall some definitions and results related to thebunches of chains considered by Bondarenko in [7]. We rearrange the definitions to make them more convenient for our purpose and consider only the case of chains (not semi-chains) as we need only this one and it is technically much easier. In what follows k denotes an arbitrary field.

Definition A.7. A bunch of chains X ={I,E_{i},F_{i},∼ } is defined by
the following data:

(1) A set I of indexes.

(2) Two chains (i.e., linear ordered sets) E_{i} and F_{i} given for each
i∈I.

Put E:=S

i∈IEi, F:=S

i∈IFi and |X|:=E∪F.

(3) An equivalence relation ∼ on |X| such that each equivalence class consists of at most 2 elements.

We also write x−y if x ∈ E_{i}, y ∈ F_{i} or vice versa (with the same
index i). Moreover, we consider the ordering on |X| which is just
the union of all orderings on E_{i} and F_{i} (i.e., x < y means that x, y
belong to the same chain E_{i} or F_{i} and x < y in this chain).

If a bunch of chains X = {I,E_{i},F_{i},∼ } is given, define the corre-
sponding k-category C = C(X) and the corresponding C-bimodule
U=U(X) as follows:

• The objects of C are the equivalence classes of |X| with respect to ∼.

18

• If a, b are two such equivalence classes, a basis of the morphism
space C(a, b) consists of elements p_{xy} with y∈a, x∈b, y <

x and, if a=b, the identity morphism 1_{a}.

• The multiplication is given by the rule: p_{xy}p_{yz} = p_{xz} if z <

y < x, while all other possible products are zeros.

• A basis of U(a, b) consists of elements u_{xy} with x ∈ b∩E,
y∈a∩F and x−y.

• The action of C on U is given by the rule: p_{zx}u_{xy} = u_{zy} if
x < z; u_{xy}p_{yt} =u_{xt} if t < y, while all other possible products
are zeros.

We also consider theadditive hull C^{∞} of the category C and the nat-
ural prolongation of the bimodule U onto C^{∞}, which we also denote
by U.

Thecategory of representations of the bunch X is then defined as the category El(U) of the elements of this bimodule [9]. In other words, the objects of El(U) are the elements of S

AEl(A, A) where A runs
through the objects of C^{∞}. A morphism u→u^{0}, where u∈El(A, A) ,
u^{0} ∈El(A^{0}, A^{0}) , is a morphism α :A →A^{0} such that αu =u^{0}α. One
can easily verify that this definition gives just the same representations
as the definition from [7]. Note that in [7] a more general situation was
investigated, but we only need this case, which is essentially simpler
than the general one. The following result is the specialization of the
description of the representations given in [7] to our case, though it can
be obtained directly using the same recursive procedure. First define
some combinatorial objects called “strings” and “bands.”

Definition A.8. Let X={I,E_{i},F_{i},∼ } be a bunch of chains.

(1) An X-word is a sequence w=x_{1}r_{1}x_{2}r_{3}x_{3}. . . r_{m}x_{m}, where x_{k} ∈

|X| and each r_{k} is either ∼ or −, such that:

(a) xk−1r_{k}x_{k} in |X|;

(b) if r_{k} =∼, then r_{k+1} =− and vice versa.

Possibly m = 1 , i.e., w=x for some x∈ |X|.

(2) Call an X-word w as above an X-cycle if r_{2} = r_{m} =∼ and
x_{m}−x_{1}. (Note that in this case m is always even.)

(3) Call an X-word fullif, whenever x_{1} is not a unique element in
its equivalence class, then r_{2} =∼ and, whenever x_{m} is not a
unique element in its equivalence class, then r_{m} =∼.

(4) Call an X-cycle w= x_{1}r_{2}x_{2}. . . r_{m}x_{m} aperiodic if it cannot be
written as a repetition of a shorter cycle v: w6=vrvr . . . v for
any r∈ { −,∼ }.

(5) We say that an equivalence class a occurs in a word w if w contains a subword x in case a ={x} is a singleton, or either

19

a subword x ∼ y or a subword y ∼ x in case a = {x, y} with x 6= y. In the former case we say that this occurrence corresponds to the occurrence of x, while in the latter case we say that it corresponds to both the occurrence of x and to the occurrence of y. Denote by ν(a, w) the number of occurrences of a in w.

Definition A.9. For an X-word w = x_{1}r_{2}x_{2}. . . r_{m}x_{m} call its ∼-
subword any subword of the form v =x∼y as well as that of the form
v = x if x ∈ X is unique in its equivalence class. In the latter case
put |v|={x}, while in the former case put |v|={x, y}. Denote by
[w] the collection of all ∼-subwords of w.

Note that if w is a cycle, it contains no entries x ∈ |X| such that x is unique in its equivalence class.

Definition A.10. For any full X-word w = x_{1}r_{2}x_{2}. . . r_{m}x_{m}, define
the corresponding string representation u=u_{s}(w) of the bunch X as
follows:

(1) u∈U(A, A) where A=L

v∈[w]|v|.

(2) Suppose there is a subword v_{1} −v_{2} in w with v_{i} ∈ [w] . Let
x be the right end of the word v_{1} and y be the left end of the
word v2. Then U(A, A) has a direct summand U(|v1|,|v2|)⊕
U(|v2|,|v1|) and we define the corresponding components of u
to be (0, u_{xy}) if x∈E and (u_{yx},0) if x∈F.

(3) All other components of u are defined to be zero.

Definition A.11. For any pair (w, π(t)) where w is an aperiodic X-
cycle and π(t) 6= t^{d} is a primary polynomial over k (i.e., a power
of an irreducible one), define the corresponding band representation
u=ub(w, π(t)) of the bunch X as follows:

(1) u∈U(A, A) where A=L

v∈[w]d|v| and d= degπ(t) .

(2) Suppose there is a subword v_{1} −v_{2} in w with v_{i} ∈ [w] . Let
x be the right end of the word v_{1} and y be the left end of the
word v_{2}. ThenU(A, A) has a direct summand

U(d|v_{1}|, d|v_{2}|)⊕U(d|v_{2}|, d|v_{1}|)'

Mat(d×d,U(|v_{1}|,|v_{2}|)⊕Mat(d×d,U(|v_{2}|,|v_{1}|)))

and we define the corresponding components of u to be (0, u_{xy}I)
if x∈E and (uyxI,0) if x∈F, where I denotes the unit d×d
matrix.

(3) Let now v_{1} be the last and v_{2} be the first ∼-subword in w
(they may coincide), x be the right end of the word v_{1} and

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y be the left end of the word v2. Then U(A, A) has a direct summand

U(d|v_{1}|, d|v_{2}|)⊕U(d|v_{2}|, d|v_{1}|)'

Mat(d×d,U(|v_{1}|,|v_{2}|)⊕Mat(d×d,U(|v_{2}|,|v_{1}|)))

and we define the corresponding components of u to be (0, u_{xy}J)
if x∈E and (u_{yx}Φ,0) if x∈F, where Φ denotes the Frobe-
nius matrix with the characteristic polynomial π(t) .

(4) All other components of u are defined to be zero.

Theorem A.12. (1) All representations us(w) and ub(w, π(t)) de- fined above are indecomposable and each indecomposable repre- sentation of X is isomorphic to one of these representations.

(2) The only possible isomorphisms between these representations are the following:

(a) u_{s}(w) ' u_{s}(w^{∗}) if w = x_{1}r_{2}x_{2}. . . r_{m}x_{m} and w^{∗} = x_{m}r_{m}
xm−1. . . r_{1}x_{0}, the reversed word.

(b) u_{b}(w, π(t)) ' u_{b}(w^{0}, π^{0}(t)) if w = x_{1}r_{2}x_{2}. . . r_{m}x_{m}, w^{0} =
x_{2k+1}r_{2k+2}x_{2k+2}. . . r_{2k}x_{2k} is a cyclic permutation of wand
π^{0}(t) = π(t) for k even, while for k odd π^{0}(t) = t^{d}π(1/t).
(c) u_{b}(w, π(t)) ' u_{b}(w^{0}, π^{0}(t)) if w = x_{1}r_{2}x_{2}. . . r_{m}x_{m}, w^{0} =
x_{2k}r_{2k}x2k−1. . . r_{2k+2}x_{2k+1} is a cyclic permutation of the re-
versed word and π^{0}(t) = π(t) for k odd, while for k even
π^{0}(t) = t^{d}π(1/t).

(d always denotes degπ.)

References

[1] Bass, H.,Finitistic dimension and a homological generalization of semiprimary rings, Trans. Amer. Math. Soc., 95 (1960), 466-488.

[2] Baues, H.-J., Quadratic functors and metastable homotopy, J. Pure and Appl.

Algebra, 91 (1994), 49–107.

[3] Baues, H.-J., Homotopy Type and Homology, Oxford University Press, 1996.

[4] Baues, H.-J., and Drozd, Y., Representation theory of homotopy types with at most two non-trivial homotopy groups, Math. Proc. Cambridge Phil. Soc., 128 (2000), 283–300.

[5] Baues, H.-J., and Hennes, M.,The homotopy classification of (n−1)-connected (n+ 3)-dimensional polyhedra, n≥4 , Topology, 30 (1991), 373–408.

[6] Baues, H.-J., and Pirashvili, T., Quadratic endofunctors on the category of groups, Preprint MPI 95-55, Max-Plank-Institut f¨ur Mathematik, Bonn, 1995.

[7] Bondarenko, V., Representations of bundles of semichained sets and their
applications, Algebra i Analiz, 3, n^{◦}5 (1991), 38–61 (English translation:

St. Petersburg Math. J., 3 (1992), 973–996).

[8] Drozd, Y., Ad`eles and integral representations, Izvestia Acad. Sci. USSR, 33 (1969), 1080–1088. (English translation: Math. USSR Izvestija, 3 (1969), 1019–

1026.)

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